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A Method for Evaluating the Distribution of Power in a Committee System

  • L. S. Shapley (a1) and Martin Shubik (a1)


In the following paper we offer a method for the a priori evaluation of the division of power among the various bodies and members of a legislature or committee system. The method is based on a technique of the mathematical theory of games, applied to what are known there as “simple games” and “weighted majority games.” We apply it here to a number of illustrative cases, including the United States Congress, and discuss some of its formal properties.

The designing of the size and type of a legislative body is a process that may continue for many years, with frequent revisions and modifications aimed at reflecting changes in the social structure of the country; we may cite the role of the House of Lords in England as an example. The effect of a revision usually cannot be gauged in advance except in the roughest terms; it can easily happen that the mathematical structure of a voting system conceals a bias in power distribution unsuspected and unintended by the authors of the revision. How, for example, is one to predict the degree of protection which a proposed system affords to minority interests? Can a consistent criterion for “fair representation” be found? It is difficult even to describe the net effect of a double representation system such as is found in the U. S. Congress (i.e., by states and by population), without attempting to deduce it a priori. The method of measuring “power” which we present in this paper is intended as a first step in the attack on these problems.



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1 See von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior (Princeton, 1944, 1947, 1953), pp. 420 ff.

2 See Arrow, K. J., Social Choice and Individual Values (New York, 1951), p. 7.

3 For a brief discussion of some of the factors in stock voting see Gothman, H. G. and Dougall, H. E., Corporate Financial Policy (New York, 1948), pp. 5661.

4 More generally, a minimal winning coalition.

5 In the formal sense described above.

6 This statement can be put into numerical form without difficulty, to give a quantitative description of the “efficiency” of a legislature.

7 The mathematical formulation and proof are given in Shapley, L. S., “A Value for N-Person Games”, Annals of Mathematics Study No. 28 (Princeton, 1953), pp. 307–17. Briefly stated, any alternative imputation scheme would conflict with either symmetry (equal power indices for members in equal positions under the rules) or additivity (power distribution in a committee system composed of two strictly independent parts the same as the power distributions obtained by evaluating the parts separately).

8 As a general rule, if one component of a committee system (in which approval of all components is required) is made less “efficient”—i.e., more susceptible to blocking maneuvers—then its share of the total power will increase.

9 In the general case the proportion is N − M + 1: M, where M stands for the number of councilmen required for passage.

10 If there are two or more large interests, the power distribution depends in a fairly complicated way on the sizes of the large interests. Generally speaking, however, the small holders are better off than in the previous case. If there are two big interests, equal in size, then the small holders actually have an advantage over the large holders, on a power per share basis. This suggests that such a situation is highly unstable.

A Method for Evaluating the Distribution of Power in a Committee System

  • L. S. Shapley (a1) and Martin Shubik (a1)


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